Abstract
In this paper, first we show that the variance used in the Markowitz’s mean-variance model for the portfolio selection with its numerous modifications often does not properly present the risk of portfolio. Therefore, we propose another treating of portfolio risk as the measure of possibility to earn unacceptable low profits of portfolio and a simple mathematical formalization of this measure. In a similar way, we treat the criterion of portfolio’s return maximization as the measure of possibility to get a maximal profit. As the result, we formulate the portfolio selection problem as a bicriteria optimization task. Then, we study the properties of the developed approach using critical examples of portfolios with interval and fuzzy valued returns. The -cuts representation of fuzzy returns was used. To validate the proposed method, we compare the results we got using it with those obtained with the use of fuzzy versions of seven widely reputed methods for portfolio selection. As in our approach we deal with the bicriteria task, the three most popular methods for local criteria aggregation are compared using the known example of fuzzy portfolio consist of five assets. It is shown that the results we got using our approach to the interval and fuzzy portfolio selection reflect better the essence of this task than those obtained by widely reputed traditional methods for portfolio selection in the fuzzy setting.
Highlights
The mean-variance (M–V) model developed by Markowitz (1952) [1] made a great contribution to the portfolio selection theory, considering a return as the mean and a risk as the variance
We should deal with interval type of uncertainty. This type of uncertainty is the simplest one and commonly occurring in practice, we have found in the literature relatively few papers devoted to the portfolio selection problems in the interval setting
Based on the analysis of cited above papers, we can say that the use of fuzzy and interval representation of uncertain information available allows us to avoid some limitations of classical probabilistic approach to the portfolio selection concerned mainly with non-symmetrical distributions of asset returns that we usually meet in practical investment
Summary
The mean-variance (M–V) model developed by Markowitz (1952) [1] made a great contribution to the portfolio selection theory, considering a return as the mean and a risk as the variance. A fuzzy multiple-criteria multiple-period portfolio selection problem based on the proposed credibilistic mean-entropy model is presented in [32]. Based on the analysis of cited above papers, we can say that the use of fuzzy and (in relevant cases) interval representation of uncertain information available allows us to avoid some limitations of classical probabilistic approach to the portfolio selection concerned mainly with non-symmetrical distributions of asset returns that we usually meet in practical investment. K) the greater values of both local criteria PARisk and OOPR than those we have got in the case of uniform asset shares distribution (see portfolio G) were obtained This means that it is possible to find an optimal set of shares using an appropriate method for the solution of multiple criteria tasks
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